Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \tan 15^{\circ} $$

Step-by-Step Solution

Answer

\( \tan 15^{\circ} \approx 0.2679 \) when rounded to four decimal places.

Step 1: Convert Angle to Radians

The tangent function can be calculated using degrees or radians. Since we often use radians in calculus and trigonometry, let's convert the angle from degrees to radians first. The conversion formula is: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]. Applying this to \(15^{\circ}\), we get: \[ 15^{\circ} \times \frac{\pi}{180} = \frac{\pi}{12} \]. So, \(15^{\circ} = \frac{\pi}{12} \text{ radians} \).

Step 2: Use Tangent Angle Formula

Now, we need to find the tangent of \(\frac{\pi}{12}\). While calculators directly use degrees to find tangent, understanding can be deepened by considering \(\tan(15^{\circ})\) using tangent formulas. However, most straightforwardly, use \( \tan(\theta)\) directly. But knowing how it's derived can be insightful as it sometimes involves formulas like \( \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \).

Step 3: Calculate Tangent Value Using a Calculator

Using a scientific or graphing calculator, input \(15^{\circ}\) into the tangent function setting the calculator to degree mode, as calculations here are facilitated this way. Enter: \[ \tan(15^{\circ}) \], to obtain the desired value.

Step 4: Round the Result

Now, round the calculated value to four decimal places. Using a calculator, \( \tan(15^{\circ}) \approx 0.267949 \). Rounding \( 0.267949 \) to four decimal places gives \(0.2679\).

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. It represents the ratio of the opposite side to the adjacent side in a right triangle. This can be expressed according to the formula:

• \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

When angles are in question, like in our exercise with \(\tan(15^{\circ})\), the tangent function helps us find the proportional relationships in a triangle for that specific angle.
It's important to remember that the tangent function is periodic with a period of \(\pi\) radians. This means every \(\pi\), the tangent function repeats its values, creating a predictable pattern.
Moreover, the tangent of an angle approaches infinity and negative infinity as the angle approaches \((\frac{\pi}{2} + k \pi)\) for integer \(k\). Recognizing these properties aids in graphing tangent functions and solving problems in trigonometry.

Degree to Radian Conversion

Degrees and radians are two different units for measuring angles. While degrees are more intuitive (with obviously 360 degrees in a full circle), radians are used more frequently in higher mathematics because they often simplify calculations. To convert an angle from degrees to radians, we use the conversion:

• \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

This formula stems from the fact that \(180^{\circ}\) equals \(\pi\) radians, as \(360^{\circ}\) completes a circle, coinciding with \(2\pi\) radians.
In our exercise, we converted \(15^{\circ}\) to radians by using the formula, resulting in \(\frac{\pi}{12}\). Knowing how to convert between these units is crucial for solving many trigonometric problems effectively. Radians provide a natural way of describing angles, particularly when working with calculus and complex numbers.

Angle Formulas

Angle formulas in trigonometry are tools for simplifying the calculation of trigonometric functions at various angles. These include the angle addition and subtraction formulas:

• \(\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}\)

For example, while calculating \(\tan(15^{\circ})\), one might consider expressing \(15^{\circ}\) in terms of familiar angles such as \(45^{\circ} - 30^{\circ}\). However, this step can be skipped when using a calculator.Another useful angle formula deals with double angles, where \(\tan(2\theta)\) is derived. Mastery of these formulas means deeper insight into how trigonometric functions interact, which profoundly benefits solving complex trigonometric identities and equations.

Rounding Numbers

Rounding numbers is an essential skill, especially when dealing with calculated values in angles or any other measures. It ensures the results are both concise and easier to interpret. In our exercise, the goal was to round the tangent value to four decimal places.Here’s how rounding to four decimal places works:

• Identify the digit in the fourth decimal place.
• Check the next digit (fifth decimal place). If it's 5 or more, we increase the fourth decimal digit by one.

In the case of \(\tan(15^{\circ}) \approx 0.267949\), we examine the number after the fourth place (which is 9). As it is greater than 5, rounding up \(0.2679\) ensures precision without compromising significant data.
Rounding helps communicate data clearly, often necessary when precision in scientific contexts must balance with clarity and simplicity for understanding.